I?. B. Cassel and R . H. Wood
6924
eats of Mixing Aqueous Electrolytes. XI. The Charge-Asymmetric Limiting Law at Low Concentrations. Barium Chloride with Sodium Chloride and Sodium Sulfate with Sodium hloride assel and H. H. Wood” Llepartnient of Chemistry, University of Dela ware, Newark, Delaware 7971 1 (Received September 28, 1973;Revised Manuscript Received June 2 1, 1974)
The heats of mixing solutions of barium chloride with sodium chloride and sodium sulfate with sodium chloride have been measured in water a t 25’ at ionic strengths as low as I = 0.02 to 0.05 mol kg-l. The solutions to be mixed have the same molal ionic strength ( I ) .The data are a test of Friedman’s limiting law for charged-asymmetric mixtures. The results show that the heats of mixing do change sharply below I = 0.2 mal kg-I in the direction required by the limiting law. The results did not go to low enough concentrations to confirm the numerical value of the limiting law slope. In both systems the asymmetry of the heat of mixing curve us. ionic strength fraction increased sharply a t very low concentrations.
Introduction The purpose of this paper is to present the results of a test of Friedman’s limiting law for charge-asymmetric ~ a mixtures. 1-3 Previous results in this l a b ~ r a t o r y ”showed trend toward higher positive values for the heat of mixing alkaline earth halides with alkali halides as the concentration was decreased to 0.5 I. The results of Smith5 working with Young for NazS04 with NaCl showed a trend toward more negative values a t low concentration contrary to the expectations of the limiting law. The present results extend the measurements on the two systems to an order of magnitude lower concentrations.
Experimental Section Sodium chloride and sodium sulfate solutions were made up from Fisher certified ACS grade salts having total reported impurities of less than 0.01%. Barium chloride solutions were prcpared from optronic grade ultrapure salt obtained from Alfa lnorganics specified to be 99.995% pure. Solutions were prepaped by weight and checked by potentiometric titration or by conductivity and found to be within 0.1% of the calculated concentration. T o double check certain unexpected results, duplicate solutions were prepared and assayed independently. ‘The calorimeter, SI commercially available LKB batch has been described e l ~ e w h e r eBecause .~ the purpose of this inivestigation is to explore low concentration limiting law behavior in experiments where heat effects will be very small, it was necessary to take extra precautions to reduce instrumental errors for low heat experiments. The largest source of (systematic) error for low heat measurements derives from the relatively large amount of heat (600 pcal) which is evolved during the rotation-mixing mode of the calorimeter which i s essential to physically intermix the two solutions. This heat of intermixing has been shownG to be highly sensitive to the weights of solution in the two compartmeni s in the calorimeter cell. Since this heat of intermixing must be subtracted from the apparent heat of mixing, the blank experiment which follows the actual mixing experiment must duplicate as nearly as possible the physical intlerniixing in the original mix. This duplication can be best aswred by the following loading procedL;lre.a9 7he Journalof Physicai C h m i s r r y . Voi. 78. No. 19. 1974
Loading Procedure. Approximately 5 g of solution A10 was loaded into the 4-ml side of the mixing cell which was either dry (in which case, wtB(I1) = 0 ) or contained 0.020 f 0.003 g of the product of the previous mixing experiment run a t the same ionic strength. The cell is then rotated as during the forthcoming mix and all but approximately 0.01 f 0.001 g of the solution A which has spilled over into the 2-ml side is removed by syringe withdrawal and weighed. The net weights of component A and B in the 4-rnl side are calculated (designated wtA(I1) and wtB(lI) in Table I). Approximately 2 g of the second solution (designated wtg(1)) is weighted into the 2-g side. This side also contains 0.01 g of component A ( W t A ( 1 ) = 0.01 g) which was left on the walls. After waiting for thermal equilibrium, the chemical, electrical calibration, and blank intermixing heats are measured. When the solution is withdrawn to prepare for the next experiment, the volume withdrawn from each side of the cell will be the same as those loaded. This procedure minimizes the systematic error due to differences in the heats of intermixing.
Results and Discussion The composition of the solutions in the calorimeter before mixing together with the experimental heats are given in Table I. The heat of mixing pure solutions of components A and B is represented by the equation2 A$
= - 4 = YA(1 - YA)I2W[RTho
+
(1 - 2yJRTkzJ
(1)
where YA is the solute ionic strength fraction of component A (the component with the highest formula weight), q is the measured heat given off, IY is the weight of water (in kg) in the final solution, RTho and RThl are parameters to describe the mixing, and I is the molal ionic strength. In fitting the actual calorimetric data in Table I, eq 1 was used to calculate 41, 911, and qF (the heats of forming the initial solutions (I and 11) and the final sgolution from the pure components). The calculated heat for the actual experiment is then 9~ - qI - 911 and a least-squares procedure is used to adjust RTho and RThl to minimize the difference between experimental and calculated heats. The scatter of the experimental heat (qexpt)from the calculated
Heats of Mixrng Aqueous Electrolytes
1925
0,5 4.1613 3.927 3.914 3.848 0.01 0.01 0.01 0.01 0.2 3.764 3.726 3.715 3 ,692 0.01 0.01 0.01 0.01
0 "0
0.01 0.01 0.01 33 1 7 3.845 3.788 3.820 0 .o
0.01 0.01 0.01 0.01 2.276 2.194 2.143 2.115
2.539 2.110 2.176 2.196 0.01 0.01 0.01 0.01
0.01 0.01 0.01 0.01 2.255 2.175 2.091 2.019
2.375 2.357 2,264 2.287
0.01 0.01 0.01 0.01 2.250 2 .OS8 1.764 1.730
1.956 2,192 2.240 2.289 0.01
40.25 34.93 35.87 34.70 36.10 36.14 35.15 33.16
40.18 35.78 35.43 35.42 36.10 35.36 34.63 34.44
7.048 6.796 6.718 6329 7.431 7.186 7.021 7.021
6.997 6.918 6.727 6.753 7.364 7.209 7.094 6.984
0.01 0.01
2.035 1.978 1.989 1.968 2.096 2.174 2.084 1.862
1.943 2.007 2.008 2.034 2.156 2.078 2.026 1.932
1.858 1.805 1.968 2.080 0.01 0.01 0.01 0 '01
0.498 0.450 0.526 0.602 0.792 0.709 0.704 0.721
0.505 0.490 0.529 0.551 0,750 0.760 0.699 0.721
1.884 0.01 2.292 0 . 0 1 2.291 4 "099 2.120 0.01 3.936 2.099 0 . 0 1
0.119 0.130 0.164 0.191 0.200
0.121 0.146 0.146 0.198 0.193
0.01 0.01 0.01 3.774 3.759 3.811 3.852
0.01
0.01 0.01 0.01
0.1 4.791. 3.898 3.755 3.731 0.01 0.01 0.01 0 .O
0 .Of
0 .0 0.01 0.01 3.774 3.831 4.772 4.274
0.01
0.05
4.248 4.155 4.076 3 .788 0.01 0.01 0.01. 0.01
0.01 0 .Ol 0.04
0.01 0.01
3.873 4,113 3.755 4.246
0.01 2.341 2.247 2,089 1.967
4,114: 3.926 3 .m 0.01 0.01
0 .a1 0.01 0.01
0.02
0 .o
0.01
0.01
TABLE 11: Mixing Parameters _ l _ _ l l l l _ l _ _ l
I
BaC12(A)-NaCl(B)
0.5 0.2 0.1
0.05 0.02 Na2S0,(A)-NaCl(B) 1. O 0.5 0.2 0.1
0.05
-.
RTho, cal kgmole-2
RThi, cal kg mole-%
105.6(16)" 127.7(12) 151.0(5) 189.0(10) 302.0(40) - 30.3(25) - 30.4(17) - 32(4) - 17(2) - l(12)
o(7) 23(5) 20(14) 77 (29) 176(131.) 0 0
29(16) 90 (8) 170(40)
The 957, confidence limits of the least significant digit is given in parentheses. For example, 105.6 & 1.6 is written 105.6 (16). a
heat (qcalcd) indicates a precision of about 1%a t high heats and about 30 pcal a t low heats. The values for the parameters which best fit this data are given in Table 11. A plot of the present results together with the results of Wood and G:hamkhau* a t I = 0.5, 1,2, and 3 for mixtures of barium chloride with sodium chloride is given in Figure 1. The agreement wheire the results overlap is quite good: 103.7 f 1.0 os. 105.6 k 1.6 for the present results at I = 0.5.
1.o
4.380 4.131 0.01 0.01
0.01 0.01 4.124 4.099
4.278 4.253 4,152 4.152 0.01 0.01
0.01 0.01 0.01 0.01 4,141 4.107
0.01
4.026 3.833 3.774 0.01
0 .o
0.01 0.01 0.01 2.288 2.257 2.165
2.406 2.323 2.472
0.01 0.01 0.01 3.721 2.035 3.769 1.992
0.01
2.375 2.209 2.475 0.01 2.301 0.01 0.02
0.5
0.2
0.01
0.01
0.01 0.01 3.830 3.786 3.792
0.01 0.01
2.044 2.234 2.286 2.007
0.01 2.306 0.01 2.143 0.01
0.01
4.173 3.823 3.753 3.744 3.734 0.01 0.01 0.01 4.317
0 .o
0.05
0.01 0.01 0.01 0.01
11,09
-10.19 10 , 39 -- 10 .e9
--10.82
-44.17
-42.07 -10.12 - 10.72 -10.79 - 10.99 -10.86 -10.32
0.01
-1.340
0.01 0.01
-1.383 -1.246
2.456 2.004 0.01 0.01
-0.560
.-
1.538 2.290 2.315 0.0 2.260 0 . 0 1 0.01 2.165 3,720 2.347 0 . 0 1 3.728 2.221 0 . 0 1 4.338 1.516 0 . 0 1 1.542 0.01 0.01
0.01 0.01
-9.36
-43.8 -40.96
-2.342 -- 2.217 -2.226 - 1,366 - 1,332 -1.280
0.1 3.841 3.702 0.01 0.01
-43.26 42,27 46 ,224 -39.23 --
-2.069 .- 2.492 -- 2.249
-0.553
0.132 0.138
-0.170 -0.139 -0.180 .- 0,118 -0.217 0,184
0.200 0.166 -0,248
0.547 -0,566 0.125 0.142
-0.227 -0,155 -0.147 --0.156 -0.157 0.134 0.146 0.227 -0.231
a The weights of components A and B are designated by WA and WB, respectively. The 4-ml compartment of the calorimeter and the 2-ml compartment are designated by I1 and I, respectively. For example, in the second mixing experiment in Table I, a mixture containing 3.927 g of BaCl?solution ( I = 0.5) and 0.01 g of NaCl solution (I = 0.5) was mixed with a mixture of 0.01 g of BaCl? solution ( I = 0.5) and 2.110 g of NaCI solution !I = 0.5) absorbing 0.03493 cal. Component A is the salt with the highest formula weight. The experimental and calculated heat released by the system are designated by qerp+,and qoalcd. Thus the BaCL-NaCl mixes were endothermic.
'
Figure 2 gives a similar plot for the mixtures of sodium SUIfate with sodium chloride. The independent results of Chenll working in this laboratory and Smith5 working with Young12 are also shown in this plot. The results are in very good agreement a t the higher concentrakions. The only serious disagreement is at I = 0.2 where Smith's value is considerably lower than the present results. The estimated 95% confidence limits of the present results are shown in Figure 2 and i t is quite possible that the disagreement at 0.2 m is within the combined experimental error of the two investigations. Friedman's higher order limiting law1r2for a mixture of a 2-1 with a 1-1 electrolyte (or a 1-2 with a 1-1 electrolyte) gives the following result3
which depends only on temperature ( T ) and the following solvent properties: dielectric constant ( D ) ,density ( d ) , specific volume (V), and a dimensionless Debye-kIGckel limiting law parameter ( A = 1.1723). For water a t 25' eq 2 becomes3
RTh, = - 633 log I (cal kg molmz)
(3
The Journal o f Physical Chemistry Val. 78. No. 19 1 9 i 4
R. B. Cassel and R. H. Wood
1926
Ii
1
?
300
I
I
400
i
0
-
200
300
3
i-" K
-
N
H
I3
=z i
l
200
i
\
IO0
I
E
a
too 0
2
I
3
I Figure 1. The value of R T k for the BaCI2-NaCI mixture is plotted vs. the total molal ionic strength: 0 indicates results of Wood and
0 -1.0
indicates the present results.
-0.5
0.5
0
1.0
I-2y Figure 3. The value of AmH/[yA(l- Y ~ ) / ~ for W ]BaC12(A)-NaCI(B) is plotted vs. (1 - 2yA). The intercept at (1 - 2yA) = 0 is RTh0 and the slope is RThl (see eq 1). The original data have been corrected to what would have been observed if the initial solutions contained pure A and B rather than a mixture of A and B. The correction is quite small.
100
50
0
I
I
I
I
2
3
1J
I
R
%
--
N
H
% I
-
0
5,
Figure 2. The value of R T 7 for the Na2SQ4-NaCI mixture is plotted vs. the total molal ionic strength: A indicates the results of Smith (ref 5) and of Wu, Smith, and Young at I = 1.0 (ref 12); 0 indicates the results of Chen (ref 1 .I); and El indicates the present results.
\
I E
a
- 50
This limiting law predicts that as the concentration goes to zero &Tho will go to positive infinity with a slope us. log I of - 4 3 3 . The resulcs in Figures 1 and 2 show an extremely sharp upturn to more positive values as the concentration gets smaller, as expected from this limiting law. However, an experimental verification of the theoretical slope in eq 2 is not possible at the present date. At I = 0.07 the slopes of hoth sets of data are an order of magnitude lower than the limiting law slope. For the mixture of barium chloride with sodiurfl chloridc , the results extend to I = 0.02 and the average slope from 1 = 0.05 to 0.02 is much larger. However it is still somewhat less than half of the theoretical slope. A plot of the present results us log I indicates that a smooth curve can be dmwn through the present results with an approach to the limiting law slope below I = 0.01. Recent theoretical calculai ions for a mixture of magnesium chloride with sodium chloride show that the limiting law for the free energy corresponding to eq 2 is approached only a t very low concentrations. Even a t I = 0.01, the calculated The Joarnal o! Pi?ysisai Chemistry. Voi. 78. No. 79. 1974
-100 -1.0
I
I
-0.5
0
I,
0.5
I.O
I-2y Figure 4. The value of A,H/[yA(l NaCI(6) is plotted vs. (1 - 2 y ~ ) .
- yA)/2wj for Na25Q4(A)-
slope is only about me-fourth of the limiting law slope. The present results constitute an experimental verification of the Friedman's limiting law in the sense that this limiting law predicted the sharp upturn observed in Figures 1 and 2 and no other explanation for this sharp upturn is available. Since this limiting law has its origin in the c log c term for charge-asymmetric electrolytes2 the present results are an experimental verification of this term.
Mixing Coefficients of Electrolytes
1927
There is one feature of the present results which is totally unexpected. Figures 3 and 4 show a plot of AmH/yA(l y*)12W us. (1 - 2 y ~for ) both mixings. For this plot the experimental data were corrected to what would have been observed if the initial solutions contained pure component A and pure component B, respectively. This correction is quite small (less than 1 cal kg molw2). According to eq 1 this plot should give a straight line, with the slope of this straight line equal to RThl (the skew term) and an intercept a t 1 - 2 : ~= 0 equal to RTho. Both plots show slopes (RTh1 ) that increase sharply as the concentration decreases. In addition, the value of RThl is the same within experimental error for the two systems at the same concentration (see Table 11). At the time that these experiments were carried out Friedman hau shown that charge asymmetric mixtures can give skew terms when strong three ion interactions are present, but ,hat these interactions would give an RThl that is p r o p o r ~ i o n to ~ l the ionic strength.14 Thus, it was expected that h e contributions of these terms to RXhl should get smaller as the concentration decreases. Recently, more refined hypernetted chain calculations of Krishnan and Friedman15 indicate a skew term increasing in magnitude w.th decreasing concentration. However the sign and magnitude of their results are not in agreement with the present experimental results. Although the experiments are very difficult to carry out
a t these low concentrations because of the extreme calorimetric sensitivity needed, the authors believe that the increase in RThl is a real effect and not a systematic error in the experiments.
Acknowledgment. The authors are grateful for the support of this work by the Office of Saline Water, IJ. s. Department of the Interior. References and Notes (1) H. L. Friedman, J. Chem. Phys., 32, 1134 (1960). (2) H. L. Friedman, “ionic Solution Theory.” Interscience, New York, N. Y., 1962, pp 244-246. (3) R. H. Wood, J. D. Patton, and M. Ghamkhar, J. Phys. Chem., 73, 346 (1969). (4) R. H. Wood and M. Ghamkhar, J. Phys. Chem,. 73, 3959 (1969). (5) M. B. Smith, Ph.D. Thesis, University of Chicago, 1942. (6) LKB Produkter AB, Fack, 161 25 Bromma 1, Sweeden, Model 10700-2. (7) P. J. Reilly and R. H. Wood, J. Phys. Chem., 76,3474 (1972). (8)A. S.Levine, Ph.D. Thesis, University of Delaware, June 1971. (9) J. S. Falcone, Jr., A. S. Levine, and R. H. Wood, J. Phys. Chem., 77, 2137 (1973). (10) Solution A is the solution of salt having the higher formula weight, solution B the solution of salt having the lower formula weight. (11) H. K. Chen, M.S. Thesis, University of Delaware, Newark, Del.. June 1971. (12) Y. C. Wu, M. B. Smith, and T. F. Young, J. Phys. Chem., 69, 1868 (1965). (13) H. L. Friedman, A. Smitherman, and R. DeSantis, J. Solution Chem., 2, 59 (1973). (14) Reference 3, p 231, or see P. J. Reilly and R. W. Wood, J. Phys. Chem., 73,4292 (1969). (15) H. L. Friedman and C. V. Krishnan, 76, 1927 (1974).
Charge-As8ymnnetricMixtures of Electrolytes at Low Ionic Strength Haralld L. Friedman” and C. V. Krishnan Depar?ment of Chemistry, State University of New York, Stony Brook, New York 11790 (Received March 28, 1974) Publication costs assisted by the National-Science Foundation
The limiting laws for the mixing coefficients gl (for the excess free energy) and hl (for the enthalpy) are derived from the cluster theory of ionic solutions. Both coefficients are found to be proportional to the square root of the ionic strength I near I = 0. The HNC approximation method is applied to models for several aqueous electrolyte mixtures a t 25’ in the range of I from 0.01 to 1.3 m. The computed g1 and hl coefficients seem consistent with the limiting laws, which dominate the concentration dependence of gl and hl only a t much smaller 1.Although the calculated I dependence of these coefficients is quite striking, it is not consrstent with the hl data of Cassel and Wood. It is remarkable that the HNC calculations fail to give satisfactory self-consistency tests a t the level of the g l coefficients. This indicates the need of more powerful approximation methods for the calculation of g1 and hl.
1. Introduction
Gibbs free energy Gex when pure The change in solutions of elect,rolytes A and B, having the Same molal 1,are rjnixedtogether at given T and p can be ionic written in the form
[ I -- 2yb#)
+
[l
-
2y]’g&)
-E-
...1
(1.1)
where ,y = ,YA is the fraction of the mixture’s ionic strength due to A and W is the mass (kg) of solvent in the mixture. From eq 1.1 it is easy to derive corresponding equations for the other extensive thermodynamic variables. Thus for the enthalpy H one finds
[ l - 23!]h#)
+
[l - 2y]2h2(1)+
... 1
(1.2)
The Journal of Physicai Chemistry. Vol. 78. No. 79 19i4
